Dynamical systems in biology
Summary
Life is non-linear. This course introduces dynamical systems as a technique for modelling simple biological processes. The emphasis is on the qualitative and numerical analysis of non-linear dynamical models. Examples are taken from biology and population models.
Content
1. Dynamical systems in 1D
- Introduction to dynamical systems, ordinary first-order differential equations in 1D.
- Linear stability analysis in 1D. Population dynamics in 1D.
- Bifurcations in 1D.
- Bistable gene expression and hysteresis in 1D.
2. Dynamical systems in 2D
- Linear 2D systems, stability and classification of fixed points, phase portraits.
- Non-linear 2D systems, numerical analysis.
- Predator-Prey models
- Limit cycles and Poincaré-Bendixson theorem, oscillators.
- Forced and coupled oscillators, phase oscillators.
3. Discrete dynamical systems
- Iterative maps in 1D, Logistic map, numerical analysis.
- Period doubling and the route to chaos.
Keywords
Non-linear dynamical systems; ordinary differential equations; qualitative analysis; numerical analysis, simulations; population models; bistability and hysteresis; biological oscillators; iterative maps in 1D, chaos and fractals.
Learning Prerequisites
Required courses
Informatics 1, II, III
Analysis I, II
Linear Algebra
Numerical Analysis
Recommended courses
Physics I, II
Important concepts to start the course
Analysis and linear algebra
Basic chemical kinetics
Learning Outcomes
By the end of the course, the student must be able to:
- Explain simple dynamical models (terms and parameters)
- Analyze dynamical models in 1D
- Analyze models in 2D including phase portraits
- Characterize fixed points and their stability
- Implement simulations of models
- Identify bifurcations in 1D models
- Model biochemical reactions
- Construct simple models
- Critique simple models
Transversal skills
- Plan and carry out activities in a way which makes optimal use of available time and other resources.
- Write a scientific or technical report.
- Demonstrate the capacity for critical thinking
Teaching methods
Lectures
Exercises
Expected student activities
Learn the theoretical material in the lectures
Solve the exercises using pencil and paper, and implement numerical analysis of models
Assessment methods
Written examination and graded exercises.
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | Yes |
Resources
Bibliography
S. H. Strogatz Nonlinear dynamics and chaos (CRC Press)
J. D. Murray Mathematical BIology (Spring)
Ressources en bibliothèque
Notes/Handbook
Course notes in pdf format
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Dynamical systems in biology
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: mandatory
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Dynamical systems in biology
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional